scholarly journals Fast solver for quasi-periodic 2D-Helmholtz scattering in layered media

2021 ◽  
Author(s):  
José Pinto ◽  
Rubén Aylwin ◽  
Carlos Jerez-Hanckes
Keyword(s):  
Convergence Rates ◽  
Periodic Structures ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Well Posedness ◽  
Fast Solver ◽  
Transmission Problems ◽  
Algebraic Error ◽  
Discrete Problems ◽  
Helmholtz Transmission Problems

We present a fast spectral Galerkin scheme for the discretization of boundary integral equations arising from two-dimensional Helmholtz transmission problems in multi-layered periodic structures or gratings. Employing suitably parametrized Fourier basis and excluding cut-off frequen- cies (also known as Rayleigh-Wood frequencies), we rigorously establish the well-posedness of both continuous and discrete problems, and prove super-algebraic error convergence rates for the proposed scheme. Through several numerical examples, we confirm our findings and show performances competitive to those attained via Nystr\"om methods.

scholarly journals APPLICATION OF COLLOCATION BEM FOR AXISYMMETRIC TRANSMISSION PROBLEMS IN ELECTRO- AND MAGNETOSTATICS

2016 ◽  
Vol 21 (1) ◽  
pp. 16-34 ◽  
Author(s):  
Olga Lavrova ◽  
Viktor Polevikov
Keyword(s):  
Integral Equation ◽  
Boundary Integral Equations ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Meridian Plane ◽  
Piecewise Constant ◽  
Boundary Interpolation ◽  
Transmission Problems ◽  
Polygonal Boundary

This paper considers the numerical solution of boundary integral equations for an exterior transmission problem in a three-dimensional axisymmetric domain. The resulting potential problem is formulated in a meridian plane as the second kind integral equation for a boundary potential and the first kind integral equation for a boundary flux. The numerical method is an axisymmetric collocation with equal order approximations of the boundary unknowns on a polygonal boundary. The complete elliptic integrals of the kernels are approximated by polynomials. An asymptotic kernels behavior is analyzed for accurate numerical evaluation of integrals. A piecewise-constant midpoint collocation and a piecewise-linear nodal collocation on a circular arc and on its polygonal interpolation are used for test computations on uniform meshes. We analyze empirically the influence of the polygonal boundary interpolation to the accuracy and the convergence of the presented method. We have found that the polygonal boundary interpolation does not change the convergence behavior on the smooth boundary for the piecewise-constant and the piecewise-linear collocation.

scholarly journals Adaptive boundary-element methods for transmission problems

1997 ◽  
Vol 38 (3) ◽  
pp. 336-367 ◽  
Author(s):  
Carsten Carstensen ◽  
Ernst P. Stephan
Keyword(s):  
Boundary Element ◽  
Boundary Integral Equations ◽  
Integral Operators ◽  
Adaptive Algorithms ◽  
Boundary Integral ◽  
Boundary Element Methods ◽  
A Posteriori ◽  
Multiplicative Constant ◽  
Transmission Problems ◽  
Weakly Singular

AbstractIn this paper we present an adaptive boundary-element method for a transmission prob-lem for the Laplacian in a two-dimensional Lipschitz domain. We are concerned with an equivalent system of boundary-integral equations of the first kind (on the transmission boundary) involving weakly-singular, singular and hypersingular integral operators. For the h-version boundary-element (Galerkin) discretization we derive an a posteriori error estimate which guarantees a given bound for the error in the energy norm (up to a multiplicative constant). Then, following Eriksson and Johnson this yields an adaptive algorithm steering the mesh refinement. Numerical examples confirm that our adaptive algorithms yield automatically good triangulations and are efficient.

TRANSMISSION PROBLEMS FOR THE LAPLACE AND ELASTICITY OPERATORS: REGULARITY AND BOUNDARY INTEGRAL FORMULATION

1999 ◽  
Vol 09 (06) ◽  
pp. 855-898 ◽  
Author(s):  
SERGE NICAISE ◽  
ANNA-MARGARETE SÄNDIG
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Pseudo Differential Operator ◽  
Elastic Fields ◽  
Linear Elastic ◽  
Transmission Problems ◽  
Nonsmooth Domains ◽  
Regularity Results

This paper is devoted to some transmission problems for the Laplace and linear elasticity operators in two- and three-dimensional nonsmooth domains. We investigate the behaviour of harmonic and linear elastic fields near geometrical singularities, especially near corner points or edges where the interface intersects with the boundaries. We give a short overview about the known results for 2-D problems and add new results for 3-D problems. Numerical results for the calculation of the singular exponents in the asymptotic expansion are presented for both two- and three-dimensional problems. Some spectral properties of the corresponding parameter depending operator bundles are also given. Furthermore, we derive boundary integral equations for the solution of the transmission problems, which lead finally to "local" pseudo-differential operator equations with corresponding Steklov–Poincaré operators on the interface. We discuss their solvability and uniqueness. The above regularity results are used in order to characterize the regularity of the solutions of these integral equations.

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